A New Level of Step-by-Step Solutions in Wolfram|Alpha
September 7, 2017 — Greg Hurst, Wolfram|Alpha Math Content
In our continued efforts to make it easier for students to learn and understand math and science concepts, the Wolfram|Alpha team has been hard at work this summer expanding our step-by-step solutions. Since the school year is just beginning, we’re excited to announce some new features.
We’re continuously working to expand our list of step-by-step topics in Wolfram|Alpha; in fact, we’ve nearly doubled the number of areas covered. We also continue to add more—over 60 topics have step-by-step coverage in domains such as algebra, calculus, geometry, linear algebra, discrete math, statistics and chemistry. Be sure to check out our examples page to see more areas that have step-by-step solutions. And with the new intermediate steps feature, expect the coverage to grow over the next few months.
It’s always nice to see a Wolfram|Alpha query covered by a sea of orange step-by-step solution buttons—something you’ll be seeing a lot more frequently as we continue to expand our collection of solution topics.
A New Look and Intermediate Steps
In addition to new areas of coverage, all step-by-step topics have been improved by adding more detail through expandable intermediate steps. Let’s use local extrema of x^3–10x + 1 as an example. Right off the bat, you’ll notice the new appearance.
In this new redesign, the steps are broken into their own blocks, the hints have a new look and there’s a new type of button that gives you the ability to drill down further and see the detailed math involved in arriving at the result of the step. In the example above, steps 3, 4 and 5 have such a button.
Let’s expand step 4:
This functionality is important for keeping the step-by-step solutions readable, while still providing all relevant information. In these steps for finding extrema, it’s important to know how to find f′(x), its roots and where it doesn’t exist. If these steps were laid out in a linear fashion, it would be easy to get lost in the steps pertinent to finding the extrema. The main steps are now the outline of how to find the extrema, and the intermediate steps provided by the new button give the specific details used in each step.
It’s sometimes the case that there are multiple details one would want within a step. In cases like this, only one set of intermediate steps is shown at a time, and clicking another will replace the one currently expanded.
Intermediate steps open a new door on the types of step-by-step solutions we can provide. In the coming months, we’ll be rolling out more content that utilizes this new feature.
The World of Step-by-Step
So what are step-by-step solutions exactly? Wolfram|Alpha has pioneered step-by-step solutions for nearly 10 years, and we continue to be the industry standard. These solutions show how to get to an answer—not just what the answer is. Let’s ask for an integral. The main results shown are calculations, i.e. the answer to the query, supplementary information and even open code. But we can go further and see how one could find the answer. Let’s click the step-by-step solution button to see.
One might think these are precomputed solutions we grab from a large table, but that’s not the case. A curated table of solutions wouldn’t be feasible because there are an infinite number of math problems. Instead we start from scratch, building a stack of functionality meant to handle any query thrown at it. The Wolfram Language is the perfect language for a project like this. Under the hood we make use of the language’s full suite of mathematical capability, along with its highly expressive, symbolic paradigm.
If the Wolfram Language can compute something, can’t we just construct the step-by-step solutions by tracing through the algorithms used? In theory, yes. For example, the Wolfram Language computes most first-order derivatives similar to the way humans do—by continually using a large table of identities. Most of the time, however, there are faster and more sophisticated algorithms a human wouldn’t possibly execute by hand. When computing an integral, for example, most likely the Risch algorithm or a Mellin convolution of Meijer G-functions is being used. Instead, our step-by-step solutions take the approach a human would most likely take—that is, using heuristics to look for substitutions, integrate by parts, etc.
The use of the Wolfram Language makes it possible to aim for the highest quality step-by-step solutions. Furthermore, our solutions are reviewed by a team of PhD’s who critique the accuracy, readability and didacticism of the solutions. As a result, Wolfram|Alpha can be thought of as a high-end virtual tutor—one with in-depth explanations that don’t miss a detail, for only $5/month. Over 80 major universities (including nearly the entire Ivy League) trust our solutions enough to have site licenses for Wolfram|Alpha Pro. This automatically puts this tutor in the palm of many students’ hands.
We’re excited to see step-by-step solutions grow, and we hope you are too. As always, your feedback is appreciated as we strive to make Wolfram|Alpha even more useful for students everywhere. Let us know what areas you’d like to see step-by-step solutions for!